In school you have learnt about Consumption, how income affects consumption, and the concept of the multiplier in the circular flow of income diagram…. let’s push on that concept at the margin…

A simple economy has the following consumption function:

C = 40 + 0.9Y − 0.002Y²

where:

0 ≤ Y ≤ 200

Y is national income in £ billions, and C is consumption in £ billions.

In this simple model, the multiplier is given by:

k = 1 / (1 − MPC)

Questions:

Last week I gave you a demand function — Qd = 100 − P², defined for 0 ≤ P ≤ 10 — and asked three things: why the price range is restricted to that interval, for what values of P demand is elastic, and how to sketch the curve with its elastic and inelastic sections labelled. Here's how I'd work through it.

There are two constraints stacked here, and a good answer separates them.

The lower bound, P ≥ 0, is purely economic. We usually rule out negative prices because they imply paying customers to take the good. (You can construct exceptions — disposing of toxic waste, negative interest rates — but in standard demand analysis, P ≥ 0 is the convention.)

The upper bound, P ≤ 10, is mathematical and economic. Quantity demanded cannot be negative. Setting Qd ≥ 0 gives us:

At P = 10 exactly, Qd = 0. This is the choke price — the price at which quantity demanded collapses to zero. Above this price the modelled function predicts negative quantity, which makes no economic sense.

So the interval is doing two jobs: P ≥ 0 enforces economic plausibility on prices; P ≤ 10 enforces non-negativity on quantities. A complete answer to Q1 names both.

The obvious move is to reach for the formula you know from A-level: PED = %∆Q / %∆P. The trouble is that that formula assumes a discrete change between two specific prices. We're not being asked about elasticity between two prices. We're being asked about elasticity at any price P along a curve. That's a different kind of question, and it needs a different tool: point elasticity.

The point elasticity formula is:

It looks unfamiliar but it's really just your A-level formula with calculus folded in. Start with what you know:

What does %∆Q actually mean? It's the change in Q (Q₂ minus Q₁) divided by the original Q₁:

(The ×100 we usually multiply by cancels out on top and bottom, so we can leave it off.) The same logic gives %∆P = ∆P / P. Putting both into the elasticity formula:

Dividing by a fraction is the same as multiplying by its reciprocal:

This is arc elasticity — it works for a discrete change between two specific prices. Notice what the two factors are doing. The first, ∆Q / ∆P, captures how Q responds to a change in P — the rate of change between two points. The second, P / Q, scales that rate so the answer comes out as a clean ratio of percentages rather than absolute units.

Now imagine we make the change in price very small — shrink Q₂ − Q₁ and P₂ − P₁ down to almost nothing. As those changes get vanishingly small, two things happen. First, ∆Q and ∆P get so tiny that we give them new names: we write dQ and dP instead — same idea as ∆, just for infinitesimally small changes. Second, the ratio ∆Q / ∆P, which was the slope of a line drawn between two nearby points on the curve, becomes the slope of the tangent at a single point. That's exactly what dQ / dP means: the derivative of Q with respect to P, which is the slope of the curve at one specific price. Substituting in:

That's the point elasticity formula. It's not a new idea — it's your A-level formula with the derivative replacing the discrete ratio. The (P / Q) scaling is identical in both versions; the only thing that's changed is that ∆Q / ∆P (which needed two specific prices to calculate) has become dQ / dP (which works at any single price along the curve).

Differentiating Q = 100 − P²:

Substituting in:

Demand is elastic when |PED| > 1. Setting up the inequality:

So demand is elastic for prices above approximately 5.77, and inelastic for prices below it. At P ≈ 5.77 demand is unit-elastic (|PED| = 1). At P = 10, |PED| → ∞ — perfectly elastic as we approach the choke price. At P = 0, |PED| = 0 — perfectly inelastic at the saturation quantity.

A quick sense-check is always worth doing. At P = 8: |PED| = 128 / 36 ≈ 3.56 — elastic. At P = 3: |PED| = 18 / 91 ≈ 0.20 — inelastic. Sensible.

The demand curve with P on the vertical axis (0 to 10) and Q on the horizontal (0 to 100), running from (0, 10) at the top-left to (100, 0) at the bottom-right, with the unit-elastic point at (≈ 66.7, ≈ 5.77) marked, and the elastic and inelastic sections clearly labelled above and below.]

Key labels:

Now: why does the elastic section appear at high prices and the inelastic section at low prices? This is the part worth slowing down for, because the intuition generalises well beyond this particular demand function.

At high P, we're close to the choke. Q is small. A small absolute fall in price moves us away from the choke and produces a relatively large percentage increase in Q — because Q is starting from a tiny base. At the same time, that absolute price change is a relatively small percentage of the high price. So PED — which is (%∆Q) / (%∆P) — has a large numerator and a small denominator. Result: elastic.

At low P, we're close to the saturation quantity. Q is large. A small absolute change in price produces a relatively small percentage change in Q (large base). And that absolute change in P is a relatively large percentage of the low price. Now PED has a small numerator and a large denominator. Result: inelastic.

This isn't a quirk of Q = 100 − P². It's a general feature of downward-sloping demand curves: elasticity tends to be higher near the choke price and lower near the saturation quantity, because the ratio of percentages reverses as you move along the curve. The same logic explains why a linear demand curve has an elastic top half and an inelastic bottom half — same intuition, different functional form.

This question is a stepping-stone from descriptive economics (calculate the elasticity between two given prices) to analytical economics (characterise elasticity as a property of a function across its whole domain). The move from ∆ to d that we just made is the move you'll be asked to make again and again at undergrad level — turning a discrete formula into a continuous one. Every interview question about elasticity, marginal cost, marginal utility, or marginal revenue is asking you to make exactly this jump.

Marginal gains… Answer for today’s question is out in two days… Come back then or subscribe to get it direct to your inbox!

P.S.

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